List of equations in classical mechanics

;Nomenclature

: "a" = acceleration (m/s²): "g" = gravitational field strength/acceleration in free-fall (m/s²): "F" = force (N = kg m/s²): "E"k = kinetic energy (J = kg m²/s²): "E"p = potential energy (J = kg m²/s²): "m" = mass (kg): "p" = momentum (kg m/s): "s" = displacement (m): "R" = radius (m): "t" = time (s): "v" = velocity (m/s): "v"0 = velocity at time t=0: "W" = work (J = kg m²/s²): "τ" = torque (m N, not J) (torque is the rotational form of force): "s"(t) = position at time t: "s"0 = position at time t=0: "r"unit = unit vector pointing from the origin in polar coordinates: "θ"unit = unit vector pointing in the direction of increasing values of theta in polar coordinates

Note: All quantities in bold represent vectors.

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [Harvnb|Mayer|Sussman|Wisdom|2001|p=xiii] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [Harvnb|Berkshire|Kibble|2004|p=1] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space. [Harvnb|Berkshire|Kibble|2004|p=2]

Classical mechanics utilises many equation—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equationss, manifolds, Lie groups, and ergodic theory. [Harvnb|Arnold|1989|p=v] This page gives a summary of the most important of these.

Equations

Velocity

: mathbf{v}_{mbox{average = {Delta mathbf{d} over Delta t}: mathbf{v} = {dmathbf{s} over dt}

Acceleration

: mathbf{a}_{mbox{average = frac{Deltamathbf{v{Delta t} : mathbf{a} = frac{dmathbf{v{dt} = frac{d^2mathbf{s{dt^2}

*Centripetal Acceleration

: |mathbf{a}_c | = omega^2 R = v^2 / R ("R" = radius of the circle, ω = "v/R" angular velocity)

Momentum

: mathbf{p} = mmathbf{v}

Force

: sum mathbf{F} = frac{dmathbf{p{dt} = frac{d(mmathbf{v})}{dt}

: sum mathbf{F} = mmathbf{a} quad (Constant Mass)

Impulse

: mathbf{J} = Delta mathbf{p} = int mathbf{F} dt :

mathbf{J} = mathbf{F} Delta t quad
if F is constant

Moment of inertia

For a single axis of rotation:The moment of inertia for an object is the sum of the products of the mass element and the square of their distances from the axis of rotation:

I = sum r_i^2 m_i =int_M r^2 mathrm{d} m = iiint_V r^2 ho(x,y,z) mathrm{d} V

Angular momentum

: |L| = mvr quad if v is perpendicular to r

Vector form:

: mathbf{L} = mathbf{r} imes mathbf{p} = mathbf{I}, omega

(Note: I can be treated like a vector if it is diagonalized first, but it is actually a 3×3 matrix - a tensor of rank-2)

r is the radius vector.

Torque

: sum oldsymbol{ au} = frac{dmathbf{L{dt} : sum oldsymbol{ au} = mathbf{r} imes mathbf{F} quad if |r| and the sine of the angle between r and p remains constant.: sum oldsymbol{ au} = mathbf{I} oldsymbol{alpha} This one is very limited, more added later. α = dω/dt

Precession

Omega is called the precession angular speed, and is defined:

: oldsymbol{Omega} = frac{wr}{Ioldsymbol{omega

(Note: w is the weight of the spinning flywheel)

Energy

for "m" as a constant:

: Delta E_k = int mathbf{F}_{mbox{net cdot dmathbf{s} = int mathbf{v} cdot dmathbf{p} = egin{matrix}frac{1}{2}end{matrix} mv^2 - egin{matrix}frac{1}{2}end{matrix} m{v_0}^2 quad

: Delta E_p = mgDelta h quad ,! in field of gravity

Central force motion

: frac{d^2}{d heta^2}left(frac{1}{mathbf{r ight) + frac{1}{mathbf{r = -frac{mumathbf{r}^2}{mathbf{l}^2}mathbf{F}(mathbf{r})

Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then calculus must be used.

:v = v_0+at ,

:s = frac {1} {2}(v_0+v) t

:s = v_0 t + frac {1} {2} a t^2

:v^2 = v_0^2 + 2 a s ,

:s = vt - frac {1} {2} a t^2

Derivation of these equation in vector format and without having is shown here These equations can be adapted for angular motion, where angular acceleration is constant:

: omega _1 = omega _0 + alpha t ,

: heta = frac{1}{2}(omega _0 + omega _1)t

: heta = omega _0 t + frac{1}{2} alpha t^2

: omega _1^2 = omega _0^2 + 2alpha heta

: heta = omega _1 t - frac{1}{2} alpha t^2

ee also

*Acoustics
*Classical mechanics
*Electromagnetism
*Mechanics
*Optics
*Thermodynamics

Notes

References

*citation|title=Mathematical Methods of Classical Mechanics|last=Arnold|first=Vladimir I.|publisher=Springer|year=1989|isbn=978-0-387-96890-2|edition=2nd
*citation|title=Classical Mechanics|last1=Berkshire|last2=Kibble|first1=Frank H.|first2=T. W. B.|edition=5th|publisher=Imperial College Press|year=2004|isbn=978-1860944352
*citation|title=Structure and Interpretation of Classical Mechanics|last1=Mayer|last2=Sussman|last3=Wisdom|first1=Meinhard E.|first2=Gerard J.|first3=Jack|publisher=MIT Press|year=2001|isbn=978-0262194556

External links

* [http://www.astro.uvic.ca/~tatum/classmechs.html Lectures on classical mechanics]
* [http://scienceworld.wolfram.com/biography/Newton.html Biography of Isaac Newton, a key contributor to classical mechanics]


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